1. The problem is to find the inverse of the function $$f(x) = \frac{8x-4}{4+5x}$$. 2. Start by replacing $$f(x)$$ with $$y$$: $$y = \frac{8x-4}{4+5x}$$ 3. Swap the roles of $$x$$ and $$y$$ to find the inverse function: $$x = \frac{8y-4}{4+5y}$$ 4. Now, solve for $$y$$ in terms of $$x$$. Multiply both sides by the denominator $$4+5y$$: $$x(4+5y) = 8y - 4$$ 5. Distribute $$x$$: $$4x + 5xy = 8y - 4$$ 6. Group terms involving $$y$$ on one side and constants on the other: $$5xy - 8y = -4 - 4x$$ 7. Factor $$y$$ on the left side: $$y(5x - 8) = -4 - 4x$$ 8. Divide both sides by $$5x - 8$$ to isolate $$y$$: $$y = \frac{-4 - 4x}{5x - 8}$$ 9. Simplify numerator: $$y = \frac{-4(1 + x)}{5x - 8}$$ Thus, the inverse function is: $$f^{-1}(x) = \frac{-4(1 + x)}{5x - 8}$$